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\bf 
MATH 280C
\rm

\bf HOMEWORK \#3, DUE APRIL 27, 2011.\rm

Please hand in the starred problems.
In the following, all random variables are defined on the probability space
$(\Omega, \mathcal F, P)$.

1.  
A Poisson process with parameter $\lambda >0$ is a stochastic process
$N=\{N(t), t\geq 0\}$ such that 
\begin{itemize}
\item $N(0)=0$,
\item for any positive integer $n$ and times  $0\leq t_1< t_2 <\ldots < t_n$, 
$(N(t_1), N(t_2)-N(t_1),..., N(t_n) - N(t_{n-1}))$
are independent,   and 
\item for each $0\leq s<t<\infty$, $N(t)-N(s)$ is a Poisson random variable with
parameter $\lambda (t-s)$.  
\end{itemize}
Note:  $N$ is a process with stationary independent (Poisson) increments.

For each $t\geq 0$, 
let $\mathcal H_t=\sigma\{ N(s) :0\leq s\leq t\}$, the smallest $\sigma $-algebra
with respect to which $N(s) $ is measurable for each $0\leq s\leq t$.
Suppose that $(\Omega , \mathcal F, P)$ is complete. Let $\mathcal N$ denote the $P$-null
sets in $\mathcal F$. For each $t\geq 0$, let
 $\mathcal G_t =\mathcal H_t\vee  \mathcal N$, the smallest $\sigma$-algebra containing $\mathcal H_t $ and $\mathcal N$
and let $\mathcal F_t =\mathcal G_{t+} $.
\hfill\break
(a) Prove that $\{N(t), {\mathcal F_t}, t\geq 0\}$  is a submartingale.
Hint: first establish this with $\mathcal G_t $ in place of $\mathcal F_t $.
\hfill\break
(b) Use the result of (a) to conclude that 
there is a modification $\tilde N$ of $N$ such that the paths of $\tilde N$ are right continuous
with finite left limits  (c\`adl\`ag) and $\{ \tilde N_t, \mathcal F_t, t\geq 0\}$ is a submartingale
where $(\Omega, \mathcal F, \{\mathcal F_t\}, P)$ satisfies the usual conditions.
(For this reason,  when one uses a Poisson process, one usually assumes that it has c\`adl\`ag paths.)

2. Suppose that $(\Omega, \mathcal F, \{\mathcal F_t\}, P)$ is a filtered probability space.
Suppose that $S$ and $T$ are stopping times on this space. Prove that
\hfill\break
(a) $S\wedge T$ and $S\vee T$ are stopping times. \hfill\break
(b) if $S\leq T$, then $\mathcal F_S\subset \mathcal F_T$.\hfill\break
Now suppose that $\{T_n\}_{n=0}^\infty $ is a  sequence of stopping times
and that $\{\mathcal F_t, t\geq 0\}$ is right continuous.
Prove that\hfill\break
(c) $\inf_n T_n $, $\sup_n T_n $, $\liminf_n T_n $ and $\limsup_n T_n$ are also stopping times. \hfill\break
(d) if $T_{n+1} \leq T_{n}$ for all $n $, then for $T=\inf_n T_n$, $\mathcal F_T=\cap_n \mathcal F_{T_n}$.

3*.
Suppose that $\{X_t, t\geq 0\}$ is a continuous adapted process defined
on a filtered probability space $(\Omega, \mathcal F, \{\mathcal F_t\}, P)$ where
$\{\mathcal F_t \}$ is right continuous. Let $A $ be a closed set in $(-\infty, \infty)$.
Prove that 
$$ T_A =\inf\{ t\geq 0: X_t \in A\}$$
is a stopping time.
\it Hint: \rm for each positive integer $n$, $A^n = \{ x: dist(x, A) <1/n\}$ is open.

4*. A standard one-dimensional Brownian motion is a real-valued stochastic process
$\{B_t, t\geq 0\}$ such that 
\begin{itemize}
\item $B(0) =0$,
\item for any positive integer $n$ and times $0\leq t_1< t_2  <\ldots < t_n$, 
$\{B(t_1), B(t_2)-B(t_1), \ldots, B(t_n)-B(t_{n-1})\}$ are independent, 
\item for each $0\leq s< t<\infty$, $B(t)-B(s)$ has a normal distribution with mean zero and variance $t-s$,
\item the paths of $B$ are continuous.
\end{itemize}
(This is another process with stationary, independent increments).

For each $t\geq 0$, 
let $\mathcal H_t=\sigma\{ B(s) :0\leq s\leq t\}$, the smallest $\sigma $-algebra
with respect to which $B(s) $ is measurable for each $0\leq s\leq t$.
Suppose that $(\Omega, \mathcal F, P)$ is complete 
and let $\mathcal N$ denote the $P$-null sets in $\mathcal F$. 
Let $\mathcal G_t =\mathcal H_t \vee \mathcal N$ and let $\mathcal F_t =\mathcal G_{t+}$ for all $t\geq 0$.
\hfill\break
(a) Prove that $\{B(t), \mathcal F_t, t\geq 0\}$ and  $\{B_t^2-t, \mathcal F_t, t\geq 0\}$  
are martingales.
Hint: first prove this with $\mathcal G_t$ in place of $\mathcal F_t$. \hfill\break
(b) Let $T=\inf\{t\geq 0: B_t >1\}$. Is $T$ a stopping time relative to $\{\mathcal F_t, t\geq 0\}$? Make sure to justify your answer.  
\hfill\break
(c) Fix $a<0<b$ and let $S=\inf\{ s\geq 0:B_s \leq a \hbox{ or } B_s \geq b\}$.
Is $S$ a stopping time relative to $\{\mathcal F_t, t\geq 0\}$? 
Use Doob's stopping theorem  
to show that $E[S]<\infty $, $E[B_S]=0$ and to compute the probability
that $B$ hits $a$ before $b$. You should also be able to compute $E[S]$.
\hfill\break
(d) (Extra credit) Prove that  
$\{M_t=\exp(cB_t - \frac{1}{2} c^2 t ), \mathcal F_t, t\geq 0\}$ is a  martingale 
for any real number $c$. (You may use results about the moment generating function for the normal distribution.) 
Prove that $\lim_{t\to\infty} M_t $ exists a.s. Can you identify this limit?
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